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Chapter 7: Problem 43

Test for symmetry and then graph each polar equation. $$r=2+3 \sin 2 \theta$$

Short Answer

Expert verified
The given polar coordinate equation \( r = 2 + 3 \sin 2 \theta \) has symmetry about the origin. This symmetry aids in visualizing and graphing the rose curve represented by this equation.

Step by step solution

01

Test for Symmetry

Check for three kinds of symmetry in polar coordinates: the symmetry about the x-axis, y-axis, and the origin. This can be done by plugging \(-\theta\) into the equation and seeing if it simplifies to the original equation or with negative r.
02

Identify the Type of Symmetry

For the given equation \(r=2+3 \sin 2 \theta \), replace \(\theta\) with \(-\theta\) to give \(r = 2 + 3 \sin -2\theta = 2 - 3 \sin 2\theta = -(2 - 3 \sin 2\theta) \). This simplifies to \(-r\). This means the graph has symmetry about the origin.
03

Graph the Polar Equation after Recognizing the Symmetry

Create a table of values for \(\theta\) and r, and then graph these points. Remember that only some points have to be graphed due to the identified symmetry. Mirror the plotted points about the origin to get the other part of the graph because of its symmetry. The equation is a type of rose curve, and the full plot can be completed by considering the symmetry.

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Most popular questions from this chapter

Use a graphing utility to graph the polar equation. $$r=\cos \frac{3}{2} \theta$$

In Exercises \(77-80,\) convert to polar form and then perform the indicated operations. Express answers in polar and rectangular form. $$ (1+i)(1-i \sqrt{3})(-\sqrt{3}+i) $$

Prove the rule for finding the quotient of two complex numbers in polar form. Begin the proof as follows, using the conjugate of the denominator's second factor: $$\frac{r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)}{r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right)}=\frac{r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)}{r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right)} \cdot \frac{\left(\cos \theta_{2}-i \sin \theta_{2}\right)}{\left(\cos \theta_{2}-i \sin \theta_{2}\right)}$$ Perform the indicated multiplications. Then use the difference formulas for sine and cosine.

Use a graphing utility to graph the polar equation. $$r=4 \sin 6 \theta$$

Explaining the Concepts Describe the test for symmetry with respect to the line \(\theta=\frac{\pi}{2}\)
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